Han Xin ordered the soldiers.

Author: Hao Jian

Emperor Gaozu Liu Bang once asked General Han Xin, "How many soldiers do you think I can take?" Han Xin gave Liu Bang an oblique look and said, "You can take 100,000 soldiers at most!" Emperor gaozu was unhappy, thinking, how dare you look down on me! "What about you?" Han Xin proudly said: "Of course I am the more the better!" Liu added three points of unhappiness to his heart and reluctantly said, "I admire the general for being so talented." Now, I have a little question for the general. With the great talent of the general, it will be easy to answer. "Han Xin said casually," Yes, yes. " Liu bang smiled cunningly and ordered a small group of soldiers to stand in a row across the wall. Liu bang ordered: "every three people stand in a row." After the team stood, the monitor came in and reported, "There are only two people in the last row." "Liu Bang also ordered:" Every five people stand in a row. " The monitor reported, "There are only three people in the last row." Liu bang also ordered: "every seven people stand in a row." The monitor reported, "There are only two people in the last row." Liu Bang turned to Han Xin and asked, "General, how many soldiers are there in this team?" Han Xin blurted out, "Twenty-three people." Liu bang was surprised and quickly increased to ten o'clock. He thought, "This man is so capable. I have to find a fault and kill him to avoid future trouble. " On the other hand, he pretended to smile and praised a few words, and then asked, "How do you calculate?" Han Xin said, "When I was young, Huang Shigong taught my grandson how to calculate. This grandson is a disciple of Guiguzi. Calculate the algorithm that contains this problem. The formula is:

The three of them lost 70 times,

Five plum blossoms are in full bloom,

Seven sons reunited in the first half of the first month,

Divide by 105. "

Liu Bang's problems can be expressed in modern language as follows:

"A positive integer, divided by 3, divided by 5, divided by 7, is 2. If this number does not exceed 100, look for this number. "

Sun Tzu's calculation gives the solution to this kind of problem: "If three numbers or three numbers have two left, it is140; Set the remaining three to sixty-three; The number of July 7th is still two, 30 sets; If the total is 233, subtract 2 10. Where there is one left in the number of three or three, then set 70; There is one left in the number of five or five, which is twenty-one; If there is one left in the number of 77, it is more than 15 160, MINUS 150. " Explaining this solution in modern language is:

First, find out the number 70 that can be divisible by 5 and 7 and 3, the number 2 1 that can be divisible by 3 and 7 and 5, and the number 1 that can be divisible by 3 and 5 and 7.

If the required number is divided by 3 and the remainder is 2, then 70× 2 = 140, 140 is a number that can be divisible by 5 and 7 as well as by 3, and the remainder is 2.

If the required number is divided by 5 and the remainder is 3, then the number 2 1× 3 = 63, 63 is a number that can be divisible by 3 and 7, and can be divided by 5, and the remainder is 3.

If the required number is divided by 7 and the remainder is 2, then the number 15×2=30, and 30 is a number that can be divisible by 3 and 5, and then divided by 7, and the remainder is 2.

In addition, 140+63+30 = 233, because both 63 and 30 can be divisible by 3, so the remainder of 233 and 140 divided by 3 is the same, and they are both residues 2. Similarly, the remainder of 233 and 63 divided by 5 is the same, both of which are 2. So 233 is a number that meets the requirements of the topic.

The least common multiple of 3, 5, and 7 is 105, so the remainder divided by 3, 5, and 7 will not change after the integer multiple of 233 plus or minus 105, so the obtained number can meet the requirements of the topic. Because the demand is only the number of a small group of soldiers, that is to say, the number of soldiers does not exceed 100, so 233 MINUS twice of 105 to get 23 is the demand.

This algorithm has many names in China, such as "Han Xin points soldiers", "Ghost Valley calculation", "partition calculation", "pipe cutting" and "psychic calculation". The title and solution are contained in Sun Tzu's Art of War, an important mathematical work in ancient China. It is generally believed that this is the work of the Three Kingdoms or the Jin Dynasty, which is nearly 500 years later than Liu Bang's life. The arithmetic formula poem is contained in Cheng Dawei's "Arithmetic Unity" in the Ming Dynasty, and the formula implied by the numbers in the poem has long been explained. Qin, a mathematician in the Song Dynasty, popularized this problem and called it "the great solution". After this solution spread to the west, it was called "Sun Tzu's theorem" or "Chinese remainder theorem". On the other hand, Han Xin was finally killed by Liu Bang's wife Lv Hou in Weiyang Palace.

Please try to solve this problem in the same way as before: